An abstract simplicial complex is a collection of nonempty finite sets with the property that for any element , if is a nonempty subset, then . An element of of cardinality is called an -simplex. An element of an element of is called a vertex. In what follows, we may occasionally identify a vertex with its corresponding singleton set ; the reader will be alerted when this is the case.
The standard -complex, denoted by , is the simplicial complex consisting of all nonempty subsets of .
1 Geometry of a simplicial complex
Let be a simplicial complex, and let be the set of vertices of . Although there is an established notion of infinite simplicial complexes, the geometrical treatment of simplicial complexes is much simpler in the finite case and so for now we will assume that is a finite set of cardinality .
We introduce the vector space of formal –linear combinations of elements of ; i.e.,
and the vector space operations are defined by formal addition and scalar multiplication. Note that we may regard each vertex in as a one-term formal sum, and thus as a point in .
The geometric realization of , denoted , is the subset of consisting of the union, over all , of the convex hull of . If we fix a bijection , then the vector space is isomorphic to the Euclidean vector space via , and the set inherits a metric from making it into a metric space and topological space. The isometry class of is independent of the choice of the bijection .
has , so its realization is a subset of , consisting of all points on the hyperplane that are inside or on the boundary of the first octant. These points form a triangle in with one face, three edges, and three vertices (for example, the convex hull of is the edge of this triangle that lies in the –plane).
Similarly, the realization of the standard –simplex is an –dimensional tetrahedron contained inside .
A triangle without interior (a “wire frame” triangle) can be geometrically realized by starting from the simplicial complex .
Notice that, under this procedure, an element of of cardinality 1 is geometrically a vertex; an element of cardinality 2 is an edge; cardinality 3, a face; and, in general, an element of cardinality is realized as an -face inside .
In general, a triangulation of a topological space is a simplicial complex together with a homeomorphism from to .
2 Homology of a simplicial complex
In this section we define the homology and cohomology groups associated to a simplicial complex . We do so not because the homology of a simplicial complex is so intrinsically interesting in and of itself, but because the resulting homology theory is identical to the singular homology of the associated topological space , and therefore provides an accessible way to calculate the latter homology groups (and, by extension, the homology of any space admitting a triangulation by ).
As before, let be a simplicial complex, and let be the set of vertices in . Let the chain group be the subgroup of the exterior algebra generated by all elements of the form such that and . Note that we are ignoring here the –vector space structure of ; the group under this definition is merely a free abelian group, generated by the alternating products of the above form and with the relations that are implied by the properties of the wedge product.
Define the boundary map by the formula
where the hat notation means the term under the hat is left out of the product, and extending linearly to all of . Then one checks easily that , so the collection of chain groups and boundary maps forms a chain complex . The simplicial homology and cohomology groups of are defined to be that of .
Theorem: The simplicial homology and cohomology groups of , as defined above, are canonically isomorphic to the singular homology and cohomology groups of the geometric realization of .
The proof of this theorem is considerably more difficult than what we have done to this point, requiring the techniques of barycentric subdivision and simplicial approximation, and we refer the interested reader to .
- 1 Munkres, James. Elements of Algebraic Topology, Addison–Wesley, New York, 1984.
|Date of creation||2013-03-22 12:34:46|
|Last modified on||2013-03-22 12:34:46|
|Last modified by||djao (24)|
|Defines||abstract simplicial complex|